The Instructional Tasks Matter:
Analyzing the Demand of Instructional Tasks
Facilitator’s Notes
MATERIALS:
• Facilitator’s overview of the “Instructional Tasks Matter: Analyzing the Demand of
Instructional Tasks” module
• Presentation slides with notes pages
• Mathematics Common Core State Standards (CCSS) – the Standards for Mathematical
Practice and the appropriate grade-level Standards for Mathematical Content
• Participant packet
• Participant reflection form
• Facilitator reflection form
• Chart paper and markers
EQUIPMENT NEEDED:
• LCD projector or document reader
PREPARATION FOR FACILITATION:
• Read the slides and the notes pages; then solve the task that your participants will work on.
Anticipate several different ways that participants may solve the task.
MATERIALS TO BE COPIED FOR PARTICIPANTS:
• Participant packet
• The grade-level CCSS for Mathematical Content and the CCSS for Mathematical Practice
• Participant reflection form
OVERVIEW OF THE MODULE:
Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it,
yet not all tasks afford the same levels and opportunities for student thinking. [They] are central to
students’ learning, shaping not only their opportunity to learn but also their view of the subject
matter.
Adding It Up, National Research Council, 2001, p. 335
By analyzing two tasks that are mathematically similar, teachers will begin to differentiate between
tasks that require thinking and reasoning and those that require the application of previously
learned rules and procedures.
© 2013 UNIVERSITY OF PITTSBURGH
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TOTAL TIMING OF THE MODULE IS 2 HOURS (120 minutes)
5 minutes
Slides 1 – 3
Overview of the Session
Share the rationale, goals, and activities for the module.
15 minutes
Slides 4 –
18
Compare and Contrast Two Tasks
Give participants 10 minutes to analyze the two tasks.
Ask participants to work in small groups to identify the similarities and
differences between the two tasks.
Once participants have compared the tasks and identify differences
between the tasks, then share the QUASAR research. Tell participants
that others have identified the cognitive demand of tasks in the TAG.
35 minutes
Slides 7 –
15
Analyze the two tasks through the lens of the Common Core State
Standards. Which of the Content Standards will students have to make
sense of when solving each task? Which of the Standards for
Mathematical Practice will students have to make sense of when
solving each task?
Private
Think Time:
5 minutes
Small Group
Work: 10
minutes
End this discussion by asking, “How do the differences between the
Strings Task and Apples Task impact students’ opportunities to learn
the Standards for Mathematical Content and to use the Standards for
Mathematical Practice?” This is an important question to ask.
Whole
Group
Discussion:
20 minutes
5 minutes
Analyzing the Cognitive Demand of Two Mathematical Tasks
Through the Lens of the CCSS
Several research quotes are provided on slides 14 and 15. If time is
limited, you can skip the quotes.
Slides 19 –
24
Instructional Tasks: The Cognitive Demand of Tasks Matters / The
Mathematical Task Analysis Guide
Share the research related to the cognitive demand of the task. Ask
participants to compare the indicators in the “Doing Mathematics”
category with the characteristics of cognitively demanding tasks that
they identified when analyzing the two tasks.
60 minutes
Slides 25 –
31
Compare and Contrast Sets of Tasks to Determine the Cognitive
Demand of the Tasks
Three sets of four related tasks have been provided. If time is limited,
then you may choose to discuss two of the three sets. Either A – D, E
– H or I – L may be discussed.
© 2013 UNIVERSITY OF PITTSBURGH
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PARTICIPANT REFLECTIONS:
Distribute the participant reflection form.
FOLLOWING THE FACILITATION:
•
Collect the participant reflection forms and look for patterns to determine what participants
learned, and if they have misconceptions or misunderstandings.
•
Complete the facilitator reflection form and consider points that were made clear and
aspects that might need to be touched on in future work with the teachers.
© 2013 UNIVERSITY OF PITTSBURGH
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Mathematics Glossary for
The Instructional Tasks Matter: Analyzing the Demand of Instructional Tasks
Module
Cognitive Demand – The cognitive demand is the kind and level of thinking required of students in
order to successfully engage with and solve the task. (Stein, M. K., Smith, M. S., Henningsen, M.
A., & Silver, E. A., 2000. Implementing standards-based mathematics instruction: A casebook for
professional development. New York: Teachers College Press).
Common Core State Standards (CCSS) for Content – The CCSS for Mathematical Content
define what students should understand and be able to do in their study of mathematics (Common
Core State Standards, 2010). The Content Standards stress conceptual understanding of key ideas
by continually returning to organizing principles such as place value or the properties of operations
to structure those ideas. In addition, the “sequence of topics and performances” that is outlined in a
body of mathematics standards must also respect what is known about how students learn.
Common Core State Standards (CCSS) for Mathematical Practice – The CCSS for
Mathematical Practice describe varieties of expertise that mathematics educators at all levels
should seek to develop in their students. These Practice Standards rest on important “processes
and proficiencies” with longstanding importance in mathematics education. The first of these are the
NCTM process standards of problem-solving, reasoning, proof, communication, representation, and
connections. The second are the strands of mathematical proficiency specified in the National
Research Council’s report, Adding It Up (full citation below): adaptive reasoning, strategic
competence, conceptual understanding (comprehension of mathematical concepts, operations, and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately), and productive disposition (habitual inclination to see mathematics as sensible,
useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
Doing Mathematics Tasks – Doing Mathematics Tasks are tasks that require students to engage
in the highest level of thinking and reasoning in order to complete the task. (Refer to the materials in
the module for characteristics of Doing Mathematics Tasks.)
Instructional Tasks – An instructional task in the QUASAR project was defined as a segment of
classroom activity devoted to the development of a mathematical idea.
Mathematical Task Analysis Guide (TAG) – The TAG is a compilation of the characteristics of
tasks at different levels of cognitive demand. Tasks fall into four possible categories. Doing
Mathematics Tasks and Procedures With Connections Tasks are high-level tasks while Procedures
Without Connections Tasks and Memorization Tasks are low-level tasks. The guide can serve as a
means of judging or rating whether a task permits the level of thinking and reasoning required by
the CCSS.
© 2013 UNIVERSITY OF PITTSBURGH
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QUASAR Project – (Quantitative Understanding: Amplifying Student Achievement and Reasoning)
was a national project aimed at improving mathematics instruction for students attending middle
schools in economically disadvantaged communities in ways that emphasized thinking, reasoning,
problem-solving, and the communication of mathematical ideas. The project was funded by the
Ford Foundation, directed by Edward A. Silver, and headquartered at the Learning Research and
Development Center at the University of Pittsburgh.
Memorization Tasks – Tasks that require students merely to recall information are Memorization
Tasks. These tasks require students to engage in no thinking and reasoning in order to solve the
task. (Refer to the materials in the module for characteristics of Memorization Tasks.)
Procedures With Connections Tasks – Tasks that require students to engage in thinking and
reasoning in order to solve the task are called Procedures With Connections Tasks. This type of
task often provides students with more structure or even a pathway into how to solve the task;
however, students are still required to make connections to the underlying mathematical ideas of
the task. (Refer to the materials in the module for characteristics of Procedures With Connections.)
Procedures Without Connections Tasks – Tasks that require students to engage in very little
thinking and reasoning when solving the task are called Procedures Without Connections Tasks.
Students do not have to problem-solve or make connections to mathematical ideas. (Refer to the
materials in the module for characteristics of Procedures Without Connections.)
Representations – "Representation is the process of using symbols, words, illustrations, graphs,
and charts to characterize mathematical concepts and ideas. It involves creating, interpreting, and
linking various forms of information and data displays, including those that are graphic, textual,
symbolic, three-dimensional, sketched, or simulated" (NCTM, 2003, p. 3). Representations of
mathematical ideas “serve as tools for mathematical communication, thought, and calculation,
allowing personal mathematical ideas to be externalized, shared, and preserved. They help clarify
ideas in ways that support reasoning and build understanding” (National Research Council, 2001.
J. Kilpatrick, J. Swafford, & B. Findell, [Eds.]. Adding it up: Helping children learn mathematics.
Washington, DC: National Academy Press). Connecting mathematical ideas across a variety of
representations helps build robust understanding and allows learners with different preferred
learning modalities to have access to this understanding.
© 2013 UNIVERSITY OF PITTSBURGH
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Overview of the Module:
This module is foundational to the work of understanding the new Common Core
State Standards and the design of the performance-based assessments.
Without a cognitively demanding task, students will not have an opportunity to learn
to think, reason, problem solve, and communicate mathematically. We know that
ALL students can learn challenging mathematics and that our job is to help teachers
accomplish this goal. The QUASAR project, an NSF-funded study done with five
middle schools in very diverse communities in the United States, found that regular
use of high-level tasks made a difference in student performance in mathematics.
Materials:
• Slides with note pages
• Mathematics Common Core State Standards (the Standards for Mathematical
Practice and the grade-specific Standards for Mathematical Content)
• Participant handout
• Chart paper and markers
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Directions:
• Give participants a minute to read the rationale slide.
or
• Paraphrase the rationale if desired.
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Directions:
• Read the goals and activities.
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(SAY) Analyze the two tasks, the Strings Task and the Apples Task. You will find the
tasks in your participant packet. Do not solve the tasks, but instead compare the
way the tasks are written and what the task is asking the student to think about and
do. In your participant packet you will find a table where you can make a list of
similarities and a list of differences between the two tasks.
Individual Work:
• Work individually. It is important to allow for individual think time before the
group share, so that all participants will have an opportunity to formulate their
own ideas. The individual time not only provides broader access to the activity,
but also allows opportunity for a range of ideas to be developed and
discussed.
Small Group Work:
• Discuss your responses with others at your table.
Group Discussion:
• We will discuss our responses. What are some similarities and differences
between the two tasks? Chart the similarities and differences. (See possible
responses on the next two notes pages, the notes page for each task.)
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(SAY) What are the similarities and differences among the tasks?
Possible Responses:
Similarities:
• Both tasks deal with addition.
• Both tasks deal with the addition of a two-digit number and a one-digit
number.
Differences:
• One task focuses on apples and the other works on computation without a
context.
• The Strings Task asks students to solve the task more than one way.
• The Strings Task directs students to represent the problem with a number line
or a drawing.
• The Strings Task requires students to notice patterns and relationships.
• The String Task ask students to notice a pattern.
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Probing Facilitator Questions and Possible Responses (Differences):
How do the Strings Task and the Apples Task differ from each other?
•
The Strings Task provides opportunities for students to make their thinking
public and for reasoning when they model their solution to each expression.
Whereas in the Apples Task, students are just asked for the total number of
apples.
•
The Strings ask asks students to make connections between representations
(to make a diagram and then write an equation that represents the diagram).
•
In the Strings Task, students must identify the pattern in the string of
expressions/equations.
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Directions: Take 10 minutes for these questions.
Probing Facilitator Questions and Possible Responses:
Which standards for mathematical content do students have opportunities to
make sense of when solving the Strings Task?
• Students have opportunities to work on the Number and Operations in Base Ten
standards when solving the Strings Task.
Which standards for mathematical content do they have opportunities to work on
when solving the Apples Task?
• The Apples Task is a story problem so 2.OA.2A.
Which standards for mathematical practice does each task require students to
use?
• The Apples Task does not require a mathematical model, whereas the Strings Task
does.
• The Apples Task requires students to make sense of the story problem.
• The String tasks ask students to notice the structure of the tens and ones place.
Students have to notice a pattern.
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Directions:
Read the standards. Which student work demonstrates that the student has met
the standards?
2.NBT.B.6 Did we work on this standard? Yes, we worked with three-digit numbers.
We used place value and compensation to make sense of the amounts we added.
2.NBT.B.7 Did we meet this standard? Yes, because we used concrete models.
We also worked with three-digit numbers. We decomposed and recomposed
quantities.
2.NBT.B.8 Why aren’t we working on this standard? What questions would we
have to ask in order to be working on this standard? We would need to say 48 + 10
= 58, 58 + 10 = 68, 68 + 10 = 78, etc.
2.NBT.B.9 Did we work on this standard? Explain why addition and subtraction
strategies work, using place value and the properties of operations.
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(SAY) Did we work on these standards?
Some students might skip count from 48 by ten to 100.
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(SAY) How might working with quantities subtracted from 100 or adding from a twodigit amount to 100 build students’ fluidity with solving problems eventually?
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(SAY) Which of these students might students have opportunities to work
to make sense of when solving the task?
2.NBT.B.7 Students might decompose 100 into 90 + 10 and then subtract
48 from it. They might also add on tens to 48 till they arrive at 100.
2.NBT.B.9 Students are asked to use addition and subtraction when
solving the problem and to explain why either operation can be used to
solve the task.
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Facilitator Notes:
After solving the task, we will consider the kind of situational word problem that we
solved.
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Directions:
Read the directions on the slide.
Probing Facilitator Questions and Possible Responses:
What about the design of the task requires students to construct a viable
argument?
• Students must analyze the situation for the Strings Task, whereas there is not
much to figure out when doing the Apples Task.
Does the Strings Task require students to reason abstractly and
quantitatively?
• Students do not use this practice for the Strings Task, but they might use it for
the Apples Task.
Will students have to look for and make use of structure?
•
Students have to show that they know the evenness of two even numbers or
two odd numbers because they have to write a generalization or draw
conclusions. The Apples Task does not require that students explain what an
even number is.
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Facilitator Note:
After the whole group discussion of the differences and similarities between the
two tasks, you should have a list of characteristics of high-level and low-level tasks
similar to those on the Mathematical Task Analysis Guide (TAG). Participants will
analyze the TAG later in the session and ideally they should recognize the
similarity between their list and the one produced by researchers from the
QUASAR project.
Directions:
• Ask participants the question on the slide. Participants may need time to Turn
and Talk with a partner before responding to the question.
• Solicit responses from the whole group (5 minutes).
Possible Responses:
• The differences do matter because students need the kinds of opportunities
that they get from doing the Strings Task. This task develops students’ ability
to problem-solve. They must figure out what to do, implement the plan, and
then draw conclusions about how the two equations can be equivalent. If
students never get this opportunity, they don’t know what to do when specific
numbers aren’t given.
• The Strings Task also provides opportunities for students to make public their
thinking and reasoning.
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Directions:
• Give participants a minute to read the slide.
or
• Paraphrase the quote.
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Directions:
• Give participants a minute to read the slide.
or
• Paraphrase the quote.
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(SAY) Let’s look at the cognitive demands of a few more tasks and see if we make
similar observations related to the characteristics of the demands of tasks. Are
some more challenging than others? What makes them more challenging? Are
some easier?
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(SAY) We are now focusing on the first phase in the Mathematical Task
Framework. The Framework was developed by the QUASAR Project. The study
recognized that math tasks pass through phases during lessons. The most
important phase is the first, the selection of a high-level task; without a high-level
task, it is not possible to engage students in thinking and reasoning. In addition to
the selection of high-level tasks, the QUASAR project learned that it was also
important for teachers to think about how a task plays out as a teacher sets it up in
the classroom and as students explore and discuss the task. 67% of high-level
tasks are NOT carried out the way they are intended to be carried out. Therefore, it
is important that teachers have opportunities to consider ways of maintaining the
cognitive demand of tasks during implementation.
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(SAY) The QUASAR Project identifies two types of high-level tasks—“Doing
Mathematics” and “Procedures With Connections.” The “Doing Mathematics” task is
a more open-ended task, whereas in the “Procedures With Connections” task
procedures are given to the student and they do not have to figure out how to solve
the task. Instead, they must make connections or discuss mathematical
relationships when solving the task. A “Doing Mathematics” task requires that
students engage in formulating and carrying out a plan when solving a problem.
The characteristics of these tasks are on the Mathematical Task Analysis Guide
(TAG ) in your handout. Both appear on the right-hand side of the document.
Small Group:
Allow time for participants to review the characteristics in the TAG and compare
them to the list of characteristics that they generated when comparing the Strings
Task and the Apples Task.
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Whole Group Discussion:
Ask participants to be specific about how the task aligns with the features on the
TAG.
(SAY) Do any of the characteristics on the left-hand side of the TAG describe the
Strings Task? How do the characteristics that we identified for the Strings Task
compare with those on the TAG?
Probing Facilitator Questions and Possible Responses:
Do you think the Strings Task is a “Procedures With Connections” task or a
“Doing Mathematics” task?
• We call the Strings Task a “Doing Mathematics” task. This task is the highest
level task. It requires the most thinking and reasoning from students.
What kind of task is the Apples Task?
• It is a “Procedures Without Connections” task because students do not have to
demonstrate any connections.
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(SAY) Take a minute to read the characteristics in the “Doing Mathematics”
category.
Probing Facilitator Questions and Possible Responses:
How do the characteristics that we identified when discussing the Strings
Task relate to those on the TAG?
• The task focuses on working with finding the addition equations two different
ways.
• Students must notice patterns in the list of equations. They may notice that 7 + 3
is always 10, therefore the tens place always goes up by one.
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Facilitator Notes:
Three sets of tasks are available. Choose either A-D, E-H, or I-L.
Directions:
Read the directions on the slide. Give participants time to determine the differences among the four tasks
that you have selected for teacher review. Responses can be written on the recording sheet in the
participant handouts.
Tasks A-D and Possible Responses:
Task A - Low-Level Task (Memorization Task): The numbers are friendly numbers and ones that students
often learn or memorize. Most of us can immediately arrive at the sum of the two numbers.
Task B - High–Level Task (Procedures with Connections): Students are shown two sides of a scale and
asked to make the expressions balance one another without writing the same expression on each side.
This task requires that students understand the meaning of equivalence.
Task C - High–Level Task (Procedures with Connections): Students are shown an algorithm and then the
expectation is that they will just follow the procedure. Students must know the meaning of the amounts in
each place, the ones and the tens place, in order to decompose and combine the amounts.
Task D - High–Level Task (Doing Mathematics): The prompt requires that students write a story problem
that matches the addition expression. In doing so, students will have to determine what words to use to
convey the meaning of addition or subtraction.
Tasks E-H and Possible Responses:
Task E - High–Level Task (Doing Mathematics): Students must show two different ways of solving the
two-digit subtraction problem. In order to know the ways are different from each other, the student must be
flexible when problem solving.
Task F - High–Level Task (Doing Mathematics): Students must solve each task and then compare and
contrast the tasks to determine how they are similar or how they differ from each other. When comparing
the situations, students will have to write about situations in which the whole is known and the part being
subtracted is unknown. Students may also indicate that some situations involve an action and others
involve work with static sets (the vanilla cookies and the chocolate cookies).
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Directions:
Task E-H and Possible Responses::
Task G - Low-Level Task (Procedures without Connections): The student can use mental math to
solve the problems. The student can also count back by ones or tens easily. The student has to
know the meaning of the ones and tens; however, once the student knows an algorithm, this task
does not have a high level of cognitive demand.
Task H - High-Level Task (Procedures without Connections): Students are shown an algorithm for a
subtraction problem and then they are asked to repeat the problem solving process. Although they
are asked to repeat the process, the student must be aware of the place value and have knowledge
that a three-digit number can be decomposed. The student must also keep track of what gets
combined in the end.
Task I - L and Possible Responses:
Task I High-Level Task (Low Mathematics): Once students know a rule for determining the largest
number and ordering the numbers, this task can be done with very little effort. If students are not
taught a rule and this task is used early in the study and ordering of sets, then this task will require
students to know the value of the amounts in each place.
Task J High–Level Task (Doing Mathematics): Students are not given a way of thinking about this
problem. There are 23 tens in 236 and 36 tens in 368. There is only one solution path; however, if
the student was not taught a procedure initially, this would be a “Doing Mathematics” task. It would
not be long before the level of this task would not be considered a high–level task.
Task K High-Level Task (Procedures With Connections): The student can use mental math to
solve the problems. The request for students to study the patterns in the set of problems and to
write an explanation of why one place changes continuously when other numbers in the three-digit
number do not change makes this a high-level task.
Task L Low–Level Task (Memorization): Students are shown an algorithm for a subtraction
problem and then they are asked to repeat the problem solving process. Although they are asked to
repeat the process, the student must be aware of the place value and have knowledge that a threedigit number can be decomposed. The student must also keep track of what gets combined in the
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Possible Responses :
I notice that the high-level tasks include work on Standards for Mathematical
Content and the Standards for Mathematical Practice, whereas the low-level tasks
do not engage students in using the Standards for Mathematical Practice
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Directions:
Read the directions on the slide.
Probing Facilitator Questions and Possible Responses:
Let’s look at some of the tasks we called “Doing Mathematics” tasks. Did they require that students use
the Standards for Mathematical Practice to do the tasks?
Let’s look at Task K.
• Yes, students must write about the structure of mathematics. They must know that the 10s place changes
when a 10 is added to it and that the 100s place changes when 100 is added to the 100s place.
Let’s look at Task F.
• Yes, students must know that each of these problems starts with a whole amount. They must recognize that
two other parts exist. In some cases, the part being taken away is known and in other situations, it is not
known. Students must be able to model the problems with equations and manipulatives. We will also get a
sense of whether or not students refer to the context once they write an equation (MP2). They also must
know about the structure of these kinds of problems and know that it is okay to know the part being
subtracted or to not know the part being subtracted. Students must construct an argument for why the
amounts are similar or different.
Let’s look at Task E.
• In order to create more than one solution path to this task, students must know the meaning of the quantities
and that they can decompose and recompose the amounts. Students will model with mathematics. We know
if they understand the structure of subtraction and addition as the student decomposes and recomposes the
amounts.
Let’s look at Task B.
• The student must understand the meaning of equivalence (MP7). The students’ equations serve as an
argument about equivalence. Students must create an argument for determining if the expressions are
equivalent without solving each expression (MP3). In order to do so, students must know that it is acceptable
to “juggle” the amount in one set to the other set and that this will not change the sum.
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Directions:
• Give participants a minute to read the quote
or
• Paraphrase the quote if desired.
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Directions:
• Give participants a minute to read the quote
or
• Paraphrase the quote if desired.
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Directions:
Ask participants to identify aspects of the textbook pages that have a high level of cognitive demand.
If the textbook pages do not have a high level of cognitive demand, then what questions might you
ask to increase the cognitive demand of the task?
Textbook Page 1a and 1b
The last question on page 240 is a high level of demand because it requires that students consider
the relationship between the tens and ones when considering when regrouping is needed.
Other questions to consider asking:
• What does the 1 mean when you write it above the tens place?
• Look at Problem 1 (32 + 29). Can we just write 11 as the sum and then add 30 + 20 and write 50
in the sum? Will this give us the same sum? If so, why?
Textbook Page 2a and 2b
Can you do Problem 1 two different ways? Why can you add and then subtract, or why can you
show the two sets that you have (the 16 and the 11, and take 14 from the 16, this results in 2 and
then you combine this with 11).
Other questions to consider asking:
• Tell me about the two problems in box one. Are they put together or take apart type problems?
• What tells you that the amounts are being put together? What tells you when something is being
taken away?
• Can you draw the part-part whole to show which of these you know in the situation?
• Why does one situation increase the size of the set and the other problem results in a smaller
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set?
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